Georgian Mathematical Journal
1(94), No. 4, 405-417
ON SOME GENERALIZATIONS OF THE VANDERMONDE
MATRIX AND THEIR RELATIONS WITH THE EULER
BETA-FUNCTION
I. LOMIDZE
Abstract. A multiple Vandermonde matrix which, besides the powers of variables, also contains their derivatives is introduced and an
explicit expression of its determinant is obtained. For the case of
arbitrary real powers, when the variables are positive, it is proved
that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the
one-dimensional case give the well-known formula for the Euler betafunction.
It is known from the classical problem of multiple interpolation (cf. [1],
pp. 104-106; [2], pp. 13-16) that there exists the unique polynomial PN −1
of degree N − 1 which satisfies the conditions
(k−1)
(k−1)
PN −1 (xj ) = yj
, k = 1, rj , j = 1, n,
(1)
(k−1)
where xj , yj
are given elements of an arbitrary field F ; xj =
6 xl if
n
X
6 l, rj ≥ 1 and N =
rj . Hence the determinant of the N × N
j =
j=1
matrix V (x1 , r1 ; . . . ; xn , rn ) ∈ MN (F ) is nonzero. Here by Ml (F ) with
natural l we denote the set of all l × l matrices with elements from F and
by V (x1 , r1 ; . . . ; xn , rn ) the N × N block-matrix
ƒ
‚
(2)
V (x1 , r1 ; . . . ; xn , rn ) = w1 (x1 )|w2 (x2 )| · · · |wn (xn ) ,
where wj , j = 1, n, stand for the rectangular matrices
ƒ
‚
wj (x) = (xi−1 )(k−1) 1≤i≤N .
(3)
1≤k≤rj
1991 Mathematics Subject Classification. 15A15, 15A54, 33B15.
405
c 1994 Plenum Publishing Corporation
1072-947X/94/0700-0405$12.50/0 
406
I. LOMIDZE
There are other cases when one has to deal with matrices of the form
(2). For instance, the determinant of such a matrix can be effectively used
when proving the proposition on functional independence of elements from
the complete set of invariants of operators (no matter whether they are
Hermitian or not) in an N -dimensional unitary space if these operators
have rj -dimensional eigenspaces, rj > 1 (this proposition will be published
in forthcoming paper of the author). The case rj = 1 has been investigated
in [3].
For r1 = · · · = rn = 1 we have N = n and (2) becomes the Vandermonde
matrix V = V (x1 , . . . , xn ) ∈ Mn (F )
‚
ƒn
V (x1 , 1; . . . ; xn , 1) = xji−1 1
with the determinant
Y
‚
ƒn
det xi−1
=
j
1
(xj − xk ).
(4)
1≤k<j≤n
1. In what follows N = {1, 2, . . .}, F stands for the field of real numbers
R or that of complex numbers C and we shall call (2) the (r1 , . . ., rn )-tiple
Vandermonde matrix. For r1 = · · · = rn = r this matrix is said to be the
r-tiple Vandermonde matrix and we denote it by V (x1 , . . ., xn ; r).
Theorem 1. For arbitrary r1 , r2 , . . . , rn ∈ N the (r1 , r2 , . . . , rn )-tiple
Vandermonde matrix (2) with x1 , . . ., xn ∈ F satisfies the identity
det V (x1 , r1 ; . . . ; xn , rn ) =
j −1
n rY

€Y
(k!)
j=1 k=0
Y
1≤i<k≤n
(xk − xi )rk ri .
(5)
In particular,
det V (x1 , . . . , xn ; r) =
Y
€ r−1
n
(k!)
k=0
Y
2
(xk − xi )r .
(51 )
1≤i<k≤n
Before proving Theorem 1 we need some preparation.
Lemma 1. Let n, N ∈ N, 1 ≤ n ≤ N and φjk , fi : F → F be such that
φjk ∈ C k−1 (F, F ),
k = 1, rj , j = 1, n,
fi ∈ C r−1 (F, F ), r = max{rj |j = 1, n}, i = 1, N ,
and let U ∈ MN (F ) be the square N × N block-matrix
ƒ
‚
U = u1 (x1 )| · · · |un (xn ) ,
N=
n
X
rj ,
j=1
(6)
GENERALIZATIONS OF THE VANDERMONDE MATRIX
407
where
‚
ƒ
uj (x) = (fi (x)φjk (x))(k−1) 1≤i≤N , x ∈ F.
(7)
1≤k≤rj
Then we have
rj
n Y
‚
ƒY
det U = det v1 (x1 )| · · · |vn (xn )
(φjk (xj )),
(8)
j=1 k=1
with
ƒ
‚
vj (x) = (fi (x))(k−1) 1≤i≤N .
1≤k≤rj
Proof of Lemma 1. Let us denote by [k]j the column having number k in
the matrix uj , k = 1, rj , j = 1, n. It is clear that (8) is trivial if φj1 (xj ) = 0
for some j, 1 ≤ j ≤ n. Now if φj1 (xj ) 6= 0 for each j = 1, n, then it is
sufficient to show that by elementary transformations of the columns of (7)
it is possible to reduce them to the form
ƒ
‚
(9)
k]j = (fi (x))(k−1) φjk (x) 1≤i≤N .
[k]j → [e
This will be proved by induction. The column [1]j in (7) already has the
1]j = [1]j = [fi (x)φj1 (x)]1≤i≤N , j = 1, n.
desired form (9) [e
Let us assume that the columns {[k]j |k = 1, m}, 1 ≤ m ≤ rj − 1, also
have the desired form (9) for each j, j = 1, n. If now φjk (xj ) = 0 for some
k, 1 ≤ k ≤ m, then (8) is proved. Next, if
φjk (xj ) 6= 0,
(10)
for each j, k, j = 1, n, k = 1, m, then we add the sum
m
X
q=1
m−1
X
€

 (q)
€
(m−q)
(m−q+1)
q−1
q
− φj,m+1 (x) fi (x)
Cm
− φj,m+1 (x)/φjq (x) [q]j =
Cm
q=0
(which is correctly defined according to (10)) to the column [m + 1]j written
by virtue of the Leibniz formula as
m
ƒ
‚
‚X
ƒ
(m−q)
q (q)
fi (x)φj,m+1 (x) 1≤i≤N .
[m + 1]j = (fi (x)φj,m+1 (x))(m) 1≤i≤N =
Cm
q=0
Finally we obtain
‚ (m)
ƒ
^
+ 1]j = fi (x)φjm+1 (x) 1≤i≤N . „
[m + 1]j → [m
408
I. LOMIDZE
Proof of Theorem 1. Denote by {i} P
the row with the number i, i = 1, N , in
r1
the matrix (2). By adding the sum q=1
cq {q + i − 1} with the coefficients
cq ∈ F to the row {r1 + i}, i = 1, N − r1 , we get


| w
en (xn )
e1 (x1 ) |
w
det V (x1 , r1 ; . . . ; xn , rn ) = det  — — | · · · | — —  , (11)
e1 (x1 ) |
u
en (xn )
| u
ej (x), j = 1, n, is obtained from the matrix (3) after
where the matrix w
ej (x), j = 1, n, has the
eliminating the last N − r1 rows, and the matrix u
form
ƒ
‚
(12)
u
ej (x) = (xi−1 Q(x))(k−1) 1≤i≤N −r1 .
1≤k≤r
Here
Q(x) = xr1 +
r1
X
cq xq−1
q=1
is a polynomial of degree r1 .
Note that the matrix w
e1 (x) = V (x, r1 ) is the r1 × r1 Wronski matrix of
r1 functions {fi (x) = xi−1 |i = 1, r1 } and therefore
1 −1
‚
ƒ rY
e1 (x) =
det V (x, r1 ) = det w
k!.
(13)
k=0
Combining the coefficients cq ∈ F , q = 1, N − n, in such a way that Q
has the r1 -tiple root at x = x1 , we obtain
Q
(k)
Q(x) = (x − x1 )r1 ,
(14)
(x1 ) = 0,
(15)
k = 0, r1 − 1,
and now the matrix u
e1 (x1 ) vanishes by virtue of the Leibniz
(12). Therefore

e2 (x2 ) |
|
w
e1 (x1 ) | w
det V (x1 , r1 ; . . . ; xn , rn ) = det  — — | — — | · · · |
e2 (x2 ) |
|
0
| u
formula and

w
en (xn )
— — ,
u
en (xn )
which, taking into account (13) and the Laplace theorem on determinant
expansion, gives
det V (x1 , r1 ; . . . ; xn , rn ) =
1 −1
€ rY

e,
k! det U
k=0
(16)
GENERALIZATIONS OF THE VANDERMONDE MATRIX
409
ƒ
‚
e = u
e ∈ MN −r (F ), U
e2 (x2 )| · · · |e
un (xn ) , satisfies the
where the matrix U
1
conditions of Lemma 1. Hence according to (8) and (3)
n
‚
ƒY
e = det w2 (x2 )| · · · |wn (xn )
det U
(Q(xj ))rj
j=2
and from (16) together with (2) and (14) we obtain a recurrence relation
det V (x1 , r1 ; . . . ; xn , rn ) =
n
1 −1
€ rY
Y
(xj − x1 )r1 rj det V (x2 , r2 ; . . . ; xn , rn ).
k!
k=0
(17)
j=2
The induction by n gives (5) from (17) and (13).
„
2. Let the reals αi , xj , i, j = 1, n, be given. We introduce the notation
(α)n = {αi |αi 6= αj if i =
6 j, i = 1, n}.
It is known that the generalized Vandermonde matrix
‚ αi ƒn
xj 1 = V (G) (x1 , . . . , xn ; (α)n ) ≡ V (G)
(0 < x1 < · · · < xn ; α1 < α2 < · · · < αn )
is completely positive (cf.[4], p.372), i.e., all its minors are positive.
Let now 1 ≤ n ≤ N and r1 , r2 , . . . , rn ∈ N. We shall call the matrix
‚ (G)
ƒ
V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) = w1 (x1 )| · · · |wn(G) (xn )
(18)
(G)
the (r1 , r2 , . . . , rn )-tiple generalized Vandermonde matrix. Here wj , j =
1, n, is the rectangular matrix
n
X
‚
ƒ
(G)
wj (x) = (xαi )(k−1) 1≤i≤N , (rj ≥ 1, j = 1, n,
rj = N ).
1≤k≤rj
(19)
j=1
Theorem 2. The (r1 , r2 , . . . , rn )-tiple generalized Vandermonde matrix
V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N )
(0 < x1 < · · · < xn ; α1 < α2 < · · · < αn )
is positively definite.
Lemma 2. For arbitrary simultaneously nonzero reals ci ∈ R, i = 1, N ,
αi =
6 αk if i 6= k, the function f : R → R
f (x) =
N
X
i=1
ci xαi
410
I. LOMIDZE
has at most N − 1 positive zeros counted according to their multiplicities:
f (x) =
n
Y
j=1
n
X
j=1
(x − xj )rj φ(x; (α)N ),
rj ≤ N −1; φ(x; (α)N ) 6= 0, x > 0; xj > 0, j = 1, n, 1 ≤ n ≤ N.
Proof of Lemma 2. If f has only single zeros, i.e., r1 = · · · = rN = 1,
then n = N , and the assertion of Lemma 2 follows from the inequality
det V (G) 6= 0 ([4], p.372). Assume that the assertion of Lemma 2 is true for
1 ≤ rj ≤ r, j = 1, n, and prove it for N power-summands also in the case
rj 0 = r + 1 for certain j 0 , 1 ≤ j 0 ≤ n.
Let the opposite be true: say, there exist reals ci , i = 1, N , at least one
of which is nonzero, such that
f (x) =
N
X
ci xαi =
i=1
n
X
j=1
rj ≥ N ;
n
Y
(x − xj )rj φ(x; (α)N ),
j=1
1 ≤ rj ≤ r + 1, j = 1, n,
1 ≤ n ≤ N.
Let m, 1 ≤ m ≤ n, be the number of multiple zeros of f and zeros xj ,
j = 1, n, be arranged according to the decrease of their multiplicity. Then
f satisfies the conditions
f (xj ) = 0, j = 1, n,
0
f (xj ) = · · · = f
(rj −1)
(xj ) = 0,
(20)
2 ≤ rj ≤ r + 1,
j = 1, m;
(21)
here
n−m+
m
X
j=1
rj ≥ N.
(22)
The Rolle theorem, together with (20), implies that the function fe :
R → R
fe(x) = (f (x)x−α1 )0 =
N
−1
X
i=1
ei
e
ci xα
vanishes at the points ξj , 0 < xj < ξj < xj+1 , j = 1, n − 1, i.e.,
fe(ξj ) = 0, j = 1, n − 1.
(23)
GENERALIZATIONS OF THE VANDERMONDE MATRIX
411
Moreover, from (21) it follows that
fe(xj ) = fe0 (xj ) = · · · = fe(rj −2) (xj ) = 0,
i.e., fe has the form
fe(x) =
n−1
Y
k=1
(x − ξk )
=
m
Y
j=1
j = 1, m,
e (e
(x − xj )rj −1 φ(x;
α)N ) =
e (e
PN∼ (x)φ(x;
α)n ),
(24)
all roots of the polynomial PN∼ being positive and having the multiplicity
≤ r. From (24) and (22) we find
e =n−1+
N
m
m
X
X
(rj − 1) = n − 1 +
rj − m ≥ N − 1,
j=1
j=1
but this contradicts our assumption, since the function (23) is the sum of
N − 1 power-summands. „
Proof of Theorem 2. For αi = i − 1, i = 1, N , we have
V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) = V (x1 , r1 ; . . . ; xn , rn )
and, according to (5),
det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) > 0
(0 < x1 < · · · < xn ; αi = i − 1,
i = 1, N ).
It is possible to pass to arbitrary values α1 < · · · < αN starting from
the integers αi = i − 1, i = 1, N , and changing them continuously, but
preserving the inequalities among them. In doing so, the determinant
det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) does not vanish according to Lemma 2
and therefore for all 0 < x1 < · · · < xn ; α1 < α2 < · · · < αN we have
det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) > 0.
Since each principal minor of the matrix V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N )
can be considered as a determinant of a certain (r1 , r2 , . . . , rn )-tiple generalized Vandermonde matrix, all such minors are positive. „
Corollary 1. It follows from det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) 6= 0 that
for arbitrary reals αi , αi =
6 αk if i 6= k, i, k = 1, N , there exists unique
collection of reals ci , i = 1, N − 1, such that
N
−1
X
i=1
ci xαi + xαn = PN −1 (x)φ(x; (α)N ),
412
I. LOMIDZE
where PN −1 is any given polynomial of degree N − 1 having only positive
roots, and φ(x; (α)N ) 6= 0 for x > 0.
Note that for αi ∈ N the above proposition is nothing but the Viete theorem, since it enables us to determine the coefficients at power-summands
provided we know zeros of the sum.
3. Let n, N be integers, 1 ≤ n ≤ N , and 0 < α1 < α2 < · · · <
αN , xj > 0, j = 1, n. Transform the determinant det V (G) (x1 , r1 ; . . . ;
xn , rn ; (α)N ), taking the factor αi , i = 1, N , out of the i-th row and taking
into account that
Γ(αi )
xαi −k+1 = (xαi −1 )(k−2) , k = 1, 2, . . . .
Γ(αi + 2 − k)
αi−1 (xαi )(k−1) =
Here we put
α (−q)
(x )
=
Z
|
0
x
dx · · ·
{z
q−tiple
Z
x
0
}
1
dxx =
Γ(q)
α
Z
0
x
(x − t)q−1 tα dt,
α ≥ 0, q = 1, 2, . . . ,
Z x
(xα )(−1) =
tα dt, α ≥ 0, x > 0,
(25)
0
and Γ(z) = (z−1)! is the gamma-function. Taking into account the corollary
of Theorem 2, after the appropriate transformations we get
det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) =
= (−1)σ1
N
Y
αi det V (G) (x1 , r1 − 1; . . . ; xn , rn − 1; (α − 1)N −n ) det A,
i=1
(26)
where
σ1 =
n
X
j=1
j(rj − 1), (α−1)N = {αi −1|αi 6= αj if i 6= j, i, j = 1, N },
and A ∈ Mn (F ) is a square matrix of the form
A = [aij ] =
hZ
xj
xj−1
Here we use the notation
φi (t; (α − 1)Ne )
n
Y
p=1
e =N −n+1
N
i
(t − xp )rp −1 dt .
(27)
(28)
GENERALIZATIONS OF THE VANDERMONDE MATRIX
and φi (t; (α − 1)Ne ) is defined by the formula
φi (t; (α − 1)Ne )
n
Y
p=1
N
−n
X
(t − xp )rp −1 =
ci+q tαi+q −1 , i = 1, n,
413
(29)
q=0
the coefficients ci+q being defined according to the corollary of Theorem 2.
Rearranging the columns of det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) which contain higher derivatives, we can reduce it to the form
det V (G) (x1 , r1 ; . . . ; xn , rn ; (α)N ) =
e
(30)
= (−1)σ2 det V (G) (x1 , r1 − 1; . . . ; xn , rn − 1; (α)N −n ) det A,
Pn
e ∈ Mn (F ) is written as
where σ2 = j=1 (j − 1)(rj − 1) and the matrix A
e = [e
A
aij ]n1 =
n
h€
Y
(rj −1) in
=
(x − xp )rp −1 |x=xj
φei (x; (α)Ne )
1
p=1
n
in
h
Y
= (rj − 1)!φei (xj ; (α)Ne )
(xj − xp )rp −1 .
1
p=1
p6=j
(31)
Here, by definition,
φei (x; (α)Ne )
n
Y
p=1
(x − xp )rp −1 =
N
−n
X
ci+q xαi+q , i = 1, n,
q=0
from which one can find
φei (x; (α)Ne ) = xφi (x; (α − 1)Ne ),
i = 1, n.
(32)
Equating the right-hand sides of (26) and (30) and taking into account
(31) and (32), we get
det
=
‚ R xj
n
Y
xj−1
(rj − 1)!
j=1
ƒn
€ t − xp rp −1
(xj − t)rj −1 dt 1
xj − xp
Qn
=
det[φi (xj ; (α − 1)Ne )]1n p=1 xj
φi (t; (α − 1)Ne )
N
Y
i=1
αi−1
Qn
p=1
p6=j
det V (G) (x1 , r1 − 1; . . . ; xn , rn − 1; (α)N )
.
det V (G) (x1 , r1 − 1; . . . ; xn , rn − 1; (α − 1)N )
(xp 6= xj if p 6= j; x0 = 0; i, j = 1, n).
For the right-hand side ratio Lemma 1 obviously gives
n
Y
det V (G) (x1 , r1 − 1; . . . ; xn , rn − 1; (α)N −n )
r −1
xj .
=
det V (G) (x1 , r1 − 1; . . . ; xn , rn − 1; (α − 1)N −n ) j=1 j
414
I. LOMIDZE
Hence we have the formula
€ t − xp rp −1
ƒn
(xj − t)rj −1 dt 1
xj − xp
Qn
=
r
det[φi (xj ; (α − 1)Ne )]n1 p=1 xj j
Qn
p=1 (rp − 1)!
=
QN
i=1 αi
n
X
rp = N, xp > 0, p, i, j = 1, n; x0 = 0),
(xp =
6 xj if p 6= j; rp ∈ N,
det
‚ R xj
xj−1
φi (t; (α − 1)Ne )
Qn
p=1
p6=j
p=1
which can be rewritten as
det
‚R1
xj−1 /xj
(1 − u)rj −1 φi (uxj ; (α − 1)Ne )
Qn
p=1
p6=j
1)Ne )]1n
€ xj u − xp rp −1 ƒn
du 1
xj − xp
=
det[φi (xj ; (α −
Qn
k=1 Γ(rk )
(33)
= Q
N
i=1 αi
n
X
6 j; rp ∈ N,
(xp 6= xj if p =
rp = N, xp > 0, p, i, j = 1, n; x0 = 0).
p=1
In the particular case where αi = r0 − 1 + i, r0 > 0, i = 1, N , we have
φi (t; (α − 1)Ne ) =
= tr0 +i−2
€
n
Y
n
€Y
(t − xp )rp −1
p=1
(t − xp )rp −1
p=1
−1
N
−n
X
−n
−1 NX
ci+q tr0 −1+i+q−1 =
q=0
ci+q tq = tr0 +i−2 , i = 1, N .
q=0
The latter equality is valid because, according to the Viete theorem, there
exist reals ci+q such that
N
−n
X
ci+q tq =
n
€Y
p=1
q=0

(t − xp )rp −1 .
Moreover,
Pn
Γ(r0 + p=1 rp )
Γ(r0 + N )
(r0 − 1 + i) =
αi =
=
Γ(r0 )
Γ(r0 )
i=1
i=1
N
Y
N
Y
GENERALIZATIONS OF THE VANDERMONDE MATRIX
415
and (33) transforms to the equality
‚
R1
Qn € xj u − xp rp −1 ƒn
ur0 +i−2 (1 − u)rj −1 p=1
du 1
det xi−1
j
xj −1/xj
xj − xp
p6=j
=
n
det[xi−1
j ]1
Qn
p=0 Γ(rp )
(34)
= Pn
Γ( p=0 rp )
6 j; rp ∈ N, xp > 0, p, i, j = 1, n; x0 = 0, r0 > 0).
(xp =
6 xj if p =
For the case n = 1 the formula (34) gives the well-known expression
Z
1
0
ur0 −1 (1 − u)r1 −1 du =
Γ(r0 )Γ(r1 )
= B(r0 , r1 ),
Γ(r0 + r1 )
(35)
for the Euler integral of the first kind.
Now, we introduce the notation
Bn (r) = B(r0 , r1 , . . . , rn ) =

 1,
n = 0,



n

Y
‚ i−1 R1 i−1
€ xj u−xk rk −1 ƒn Ž

n
rj −1
du 1 det[xi−1
det
x
u
(1−u)
j
j ]1 ,
=
x
−x
j
k
xj−1 /xj

k=0

(36)

k6=j


 n ≥ 1 (x 6= x if p 6= j, p, j = 0, n, x = 0)
0
p
j
for arbitrary {rj |rj > 0, j = 0, n} ≡ r.
Since the Euler formula (35) is valid for arbitrary complex r0 , r1 , Re r0 >
0, Re r1 > 0, there arises the problem:
Problem. Is the equality
Bn (r) = B(r0 , r1 , . . . , rn ) =
Qn
j=0 Γ(rj )
Pn
Γ( j=0 rj )
(37)
fulfilled for an arbitrary complex rj , j = 0, n, n ≥ 2? (Note that this case
is not covered either by (34) or by (35)).
4. Put 2 ≤ n ≤ N and 1 ≤ m ≤ min{rj |j = 1, n}. Transform the
determinant on the left-hand side of (5), taking the factor Γ(i)/Γ(i − m)
out of each row having the number i ≥ m + 1 and keeping in mind that
 Γ(i − m)
Γ(i)
xi−1
‘(k−1)
=
Γ(i − m) i−k
= (xi−m−1 )(k−m−1)
x
Γ(i − k + 1)
(i = m + 1, N , 1 ≤ m ≤ min{r|j = 1, n}, k = 1, 2, . . . ).
416
I. LOMIDZE
After easy but rather long calculations, using (25) and (5) we get the
formula
m 
n
‘
Y
Y
Γ(k)
det[B (1) | . . . |B (n) ]
=
Γ(rj − k + 1) ,
det V (x1 , . . . , xn ; m)
Γ(N − k + 1) j=0
(38)
k=1
where B (j) , j = 1, n, is a rectangular matrix of the form
(j)
B (j) = [bik ]1≤i≤mn =
h
= xi−k
j
Z
1
xj−1 /xj
(1 − u)
1≤k≤m
n
Y
€ xj u
rj −m k+i−2
u
p=0
p6=j
− xp rp −m i
du 1≤i≤mn
xj − xp
1≤k≤m
(xp 6= xj if p =
6 j; x0 > 0; j = 1, n).
(39)
In particular, for m = 1 (38) reduces to (34).
5. Let us now apply the above results to the case n = 1, N = r, x > 0.
It is easy to show that for the determinant of the matrix
‚
ƒr
V (G) (x, r; (α)r ) = (xαi )(k−1) 1 ,
which is nothing but the Wronski matrix for r functions {fi (x) = xαi |i =
1, r; x > 0}, one gets
Y
(αj − αk ),
(40)
det V (G) (x, r; (α)r ) = xβ
1≤k<j≤r
PN
where β = i=1 αi − r(r − 1)/2.
Obviously, the formula (40) generalizes (13).
Using (40), one can find values of φ(x; (α)N ) and coefficients ci for which
N
−1
X
i=1
ci xαi + xαN = (x − x1 )N −1 φ(x; (α)N ).
Namely, the Cramer formulas give
N −αi
ci = −xα
Li (αN ),
1
i = 1, N − 1,
−1
N
X
€

N
(x/x1 )αN −
φ(x; (α)N ) = (x−x1 )−(N−1) xα
(x/x1 )αi Li (αN ) , (41)
1
i=1
where
Li (α) =
N
−1
Y
p=1
p6=i
€ α − αp 
, i = 1, N − 1,
αi − αp
GENERALIZATIONS OF THE VANDERMONDE MATRIX
417
is the Lagrange elementary interpolation polynomial. (41) gives
=
φ(x1 ; (α − 1)Ne ) =
N
−1
‘
N −N 
X
xα
Γ(αN )
Γ(αi )
1
−
Li (αN ) ,
(N − 1)! Γ(αN − N + 1)
Γ(αi − N + 1)
i=1
(42)
N
−1

‘
N −N
X
xα
αN −1
αi −1
1
φ(ux1 ; (α − 1)Ne ) =
u
−
)
u
L
(α
. (43)
N
i
(u − 1)(N −1)
i=1
e = N and
Substituting (42) and (43) in (33) for n = 1, N = r1 ≥ 2, N
performing integration we obtain
−1
N
X
Γ(αi )
Γ(αn )
−
Li (αN ) =
Γ(αN − N + 1)
Γ(αi − N + 1)
i=1
= (−1)N
−1
€ NX
i=1
−1
αi−1 Li (αN ) − αN
Finally, substitution of (44) in (42) gives
φ(x1 ; (α−1)Ne ) =
N
Y
αk .
(44)
k=1
N
−1
N−N
Y
€ NX
xα
−1
1
αk .
αi−1 Li (αN )−αN
(−1)N
(N −1)!
i=1
(45)
k=1
References
1. R. W. Hamming, Numerical methods for scientists and engineers.
(Translation into Russian) Mir, Moscow, 1968; English original: McGrawHill, 1962.
2. N.A.Lebedev and V.I.Smirnov, The constructive theory of functions
of a complex variable. (Russian) Nauka, Moscow, 1964.
3. I.R.Lomidze, Criteria of unitary and orthogonal equivalence of operators. Bull. Acad. Sci. Georgia 141(1991), No. 3, 481-483.
4. F.R.Gantmakher, The theory of matrices. (Russian) Nauka, Moscow,
1988.
(Received 24.11.1992)
Author’s address:
Faculty of Physics
I. Javakhishvili Tbilisi State University
3, I. Chavchavadze Ave, 380028 Tbilisi
Republic of Georgia